The invention is based on a fuel injection system. In a fuel injection system of this kind, which has been disclosed by DE-SO 43 41 546, a first high pressure fuel reservoir is provided as a high pressure fuel source which supplies second high pressure fuel reservoirs that are each associated with a fuel injection valve. Fuel from this second high pressure fuel reservoir is supplied to the fuel injection valve by way of an on-off valve. The high pressure fuel reservoirs are respectively connected to one another by way of high pressure lines. These lines respectively provide a limited through flow cross section and consequently also have a throttling influence on the supply of fuel to the fuel injection valve. A sufficient delivery of fuel to the fuel injection valve via the respective fuel injection process depends on the size of the reservoir, its pressure level, its refilling capacity, and lastly, the throttling in the high pressure lines. If these parameters are not taken into consideration to a sufficient degree, pressure influxes can occur during the respective fuel injection process in the fuel injection valve. Increasing the parameter that produces this kind of reduction in the fuel delivery, however, leads to considerable overdimensioning. In particular, though, throttling line connections between the high pressure reservoir and the fuel injection valve also have a considerable negative influence on the assurance of a particular pressure level due to the elasticity of the lines and the resultant variable absorption volumes when there are pressure changes, even when a sufficient quantity of high pressure fuel from the high pressure reservoirs is available.